Eigensolutions of nonlinear wave equations in one dimension

Bulletin of Mathematical Biology, Vol 44, No. 3, pp. 321-337, 1982. Printed in Great Britain.

0092-82401821030321-17503.0010 Pergamon Press Ltd.

© 1982 Society for Mathematical Biology.

E I G E N S O L U T I O N S OF N O N L I N E A R W A V E E Q U A T I O N S IN O N E D I M E N S I O N

• B. F. GRAY and M. E. SHERRINGTON School of Chemistry, Macquarie University, North Ryde, N.S.W. 2113, Australia and Brighton Polytechnic, England

A class of nonlinear equations describing the steady propagation of a disturbance on the infinite interval in one dimensional space are shown under certain conditions to admit solution with a unique velocity of propagation. The class of equations describe both initial and final homogeneous steady states which are asymptotically stable with respect to uniform perturbations, in contrast to the Fisher equation, which does not.

1. I n t r o d u c t i o n . Steady, one-dimensional flame propagation has been considered by many workers over a long period of time (e.g., Frank- Kamenetskii, 1955). It was realised from a very early date that a characteristic and unique velocity of propagation of the disturbance as a fixed waveform required stringent conditions at ( - ~ ) for a wave moving from right to left to be imposed on the nonlinear function occurring in the equations. Frank-Kamenetskil (1955) points out that a single differential equation such as one derived from the Fisher equation,

D d Z ~ " - - d ~ " - (J ~ + k z ( 1 - r ) = O, (1)

cannot describe a flame-like phenomenon (here ~- would be a dimensionless temperature of the reacting medium), since the cold material into which the flame would propagate would be unstable with respect to infinitesimal perturbations. It would not wait for the flame front to arrive before beginning to react with a finite rate. This situation has not been used in flame theory to describe steady propagating flame fronts, the approach has been to insist on conditions at the upstream singularity

321

322 B . F . GRAY A N D M. E. S H E R R I N G T O N

( - ~ ) which are physically realistic and guarantee a unique wave-like solution. Thus, if we envisage a more general equation than that of Fisher,

G + 4,(z) = 0, (2)

the spatially homogeneous state into which the front propagates would be described by

dz d---t- = ~b(z) (3)

and for this to exist physically at all it is necessary that ~b'(~-)< 0 at the steady state. It seems axiomatic that steady states which are not stable or asymptotically stable in some sense will not be observed physically.

On the other hand, an extensive literature has arisen on propagating wave solutions to equations of Fisher type (see Murray, 1977), in spite of this problem. Finite domain stability can be shown for propagating wave solutions of Fisher type equations, but they are, of course, unstable on the infinite domain because of the problem of the upstream steady state.

In this paper we show conditions under which propagating wave solutions exist for a unique velocity only, for a disturbance moving towards a uniform asymptotically stable steady state in the upstream region.

2. Basic Equations and Boundary Conditions. We shall for the moment consider as our basic equations

d 2 ~ - dr - O ~ + ~b(~-, a) = 0 (4)

L d 2 a o da _ ~ - - r - dx q~(~', a ) = 0, (5)

with boundary conditions

z = 0 a = l x ~ - ~

~" = 1 a = 0 x ~ , (6)

the first derivatives also vanishing at the boundaries. These equations represent the propagation of a wave-like disturbance with constant

EIGENSOLUTIONS OF NONLINEAR WAVE EQUATIONS 323

velocity G from right to left in the x space. The transition in the medium associated with this disturbance would be schematically a ~ ' r . An obvious particular interpretation would be to take a as a dimensionless reactant concentration and ~- as a dimensionless temperature, in which case these equations represent a simple model of steady laminar adiaba- tic flame propagation, 4)0, a) being the reaction rate. In this case L would be the Lewis number, a dimensionless ratio of diffusion coefficient and thermal conductivity. The equations could also represent propagation of a general chemical transformation a ~ - , where both variables represent dimensionless concentrations. There are also many other possible interpretations of the variables relevant to the spreading of a mutant gene, as in the original Fisher problem, the propagation of an epidemic wave, the propagation of a nerve signal etc.

We shall not impose any restrictions on the analytical form of ~b(z, a) except to consider it to be a non-negative function for all 0-, a ) ~ (0, 1) × (0, 1), which is zero at least at x ~ + ~. We will also emphasise the physically realistic assumption that ~b is an increasing function of a and ~-, e.g.

0t~ 1 > 0~2~- ~ (~('TI0~I) > (~(TlOf2) ; ~ ~ ( 0 , 1). (7)

It is very convenient to consider the behaviour of equations (3) and (4) by transforming to the state plane (p, a) by elimination of the independent variable x.

Using the transformation

dT P = d x '

.we have

d p d2r dp d--~ = ~-~ = P ~--~z,

so equation (1) may be written as

dp = G - ~b0", a) d'r p (8)

By addition of (3) and (4) we have

d2~" - d2a = G dr + G da + L h 7 dx '

324 B . F . GRAY A N D M. E. S H E R R I N G T O N

and integrating and applying the boundary conditions at x =---~, we obtain the first integral of the system

d r L d a = G ( z + a - 1 ) , (9) d---x + dx

which may be written as

L d__aa = G ( z + a - 1) 1. (10) dr p

Equations (8) and (10) form the state plane equations of the problem subject to the boundary conditions

z = O p = O a = l

z = 1 p = 0 a =0. (11)

The region of physical interest consists of the 'box', ~-E [0, 1]a ~ [0, 1] and p/> 0, and a typical solution is shown in Fig. 1.

3. A Geome t r i ca l Theorem. In the analysis of the state plane behaviour of solutions of (8) and (10) we will require the following.

THEOREM 1. L e t y = y l ( x ) and y = y2(x) in tersec t at a p o i n t (Xo, Yo) and suppose that

(dy, (dy2 dx ]o > \ dx ]o"

Figure 1. A typical solution of equation (1).


Further suppose that Yl and Y2 intersect again at (xc, Yc), where

(i) Yl and Y2 have no common points between (Xo, Yo) and (Xc, y~); (ii) (dyl/dx) and (dy2/dx) exist and are continuous in that interval.

Then, at (xc, yc)

(dy, < (dy2 d x ] c ~ \ d x ] c

The proof is given in the Appendix.

4. The Case L = 1. An interesting and instructive special case, L = 1, has received particular attention in the past (Menkes, 1959; Kennerley and Adler, 1966; Murray, 1977) since it is possible to reduce our system of equations (3) and (4) to a single equation. In this case the first integral (9) may be written as

d ( z + a _ 1) = G ( z + a - 1) dx

and the only solution satisfying the boundary conditions at X = -+ ~ is the null solution

r + a = 1. (12)

Therefore, the state plane equations reduce to

d__p = G 4,(z) (13) dr p '

where q, is a function of the single variable ~-, since this uniquely defines a by (12).

Equation (13) has two singularities at z = 0 and z = 1, since q,(0)= q,(1) = 0 and p(0) = p(l) = 0.

The character of these singularities, representing the initial and final states of the transformation, is easily determined by standard techniques. The result hinges entirely on the value of d~b/dz at each of the singular points. Since G > 0 by definition, if the derivative is < 0 the singularity will be a saddle point, and if it is > 0 the singularity will be an unstable node or focus.

It is clear that for an exceptional curve to connect the two singularities

326 B.F. GRAY AND M. E. SHERRINGTON

they must both be saddle points. Also, the requirement of asymptotic stability for the initial and final steady states with respect to very long wavelength (or uniform) perturbations indicates that the physically meaningful requirement is for both singularities to be saddle points. We therefore assume this to be the case.

We are interested in the integral curve passing through r = 0, p = 0 with positive gradient. The values of G for which this curve also passes through the point ~- = 1, p = 0 will be considered 'eigenvalues' of equation (13) and the solutions, 'eigensolutions'.

The behaviour of integral curves through the initial singularity r = 0, p = 0 is shown in Fig. 2. Let us consider that at least two distinct eigenvalues G* exist, G* = G1 and G* = G2, where G1 > G: > 0.

The positive gradient of the integral curve at r = 0 is easily shown to be

G + 4 [ G 2 - 4 (-d--~-z)°] (14)

Let p = pl(r), Pz(~') denote the integral curves when G = G1, G2 respectively. Then at r = 0

\ d r ] o >0"

The negative gradient at the 'final singularity', r = 1, similarly is

(~-~Pz)~=I = G - x/[G224(d~O/dr)l] ' (15)

Figure 2.

p

I

i

T °J / The behaviour of integral curves near the initial singularity (0, 0).

"ls!x~ s~nlUAUO$!o 13u!ls!p oral J! suo!lnlOS jo Jno!Aeq~q aq, L

I 0

.0=1, ~llaelnSu!s Ie!l!u! aql le (.~)Zd pue (.~)td jo sadols aql ql!~ uo!13!pe~quo3

Salldm ! uo!l!puo3 s!ql 'maaoaq,L uo!loasaalu I ~ql ~Idde am j! 'a~Aa~OH

\~dp] \'dp]

~d :D = ~[ *p ~ (°,)rp \Zdp]

~d 'D = "[ .tp '~ (~.t j--~ - \ ' d p ]

'(£D uo!lenba Aq 'lu!od s!ql 1V • ~es '(~d''z) 'lu!od £aeu!pao amos le aauo ls¢a I le 13asaalu!

ii!~a (z)Zd pue (z)~d SaAan3 aql 'sanleAUa$!a luaaaJJ!p o~1 jo aouals.lXa aql aoj '~iaeaI 3 'pue suo!lnlos asaql jo ano!Aeqaq aql smoqs £ aan$!d

"0> \tap ]> \~dp]

aA~q aA't pu~

~D < ~O 'O > --~

'a:toj -oaaq,L "D ol loadsaJ ql!~ lua!pe~t~ (OA!ll3~OU lnq) ~ms~a~om ue s! qo!q~

LEE SNOLLVfI03 ~AVA~ ~IVHNIqNON dO SNOI,I.flqOSNHDIH


Therefore the two curves cannot cross, hence the uniqueness of the eigenvalue G*.

5. The General Case. In the investigation of the general case L # 1 we must return to equations (8) and (10) subject to the boundary conditions (11). From equation (9) we may easily show that the following in- equalities hold:

L < I r + a - l > 0

L > I " r + a - l < 0 . (16)

We will find it necessary here to restrict the discussion to the case L < 1 and so are interested in trajectories lying within the 'box' ~-E [0, 1], a E [0, 1], p > 0, containing the points (z, p, a) = (0, 0, 1) and (1, 0, 0) and lying above the plane a + z = 1. Let us denote the space (~-, p, a) as E and consider the projections of a trajectory in the planes p - z, (~'1 plane) and a - z, (7r2 plane).

For waves with L less than one, p is non-negative. A typical eigensolution is shown in Fig. 4.

If we denote the curve corresponding to an eigenvalue G* by C*, then we must be careful at this stage to distinguish between intersections of two eigensolutions C1 and Cz in E space and in intersection of the projections of C1 and C2 in one of planes ~rl or ~'z. Such an intersection would represent curves in some sense 'passing beneath ' each other in E space.

Therefore, a point D will be called a pass point in 7rl if at

'7" = "PD, P l = P 2 ----- P D , Or1 7 ~ Or2 (17)

and will be called a pass point in 7r2 if at

'7" = "rD,~ •1 = 0~2, P l :7~ P 2 , (18)

P a

r=O r=l r=O T=I

Figure 4. A typical eigensolution for L < 1.


where the subscripts l, 2 refer to the values of p and a on the curves C1 and C2 and at the point r = rD. A point D will be called an intersection point iff

'7" = 7c, P l = P 2 = PD, 0[.1 = 0l'2 ~-" OlD. (19)

To investigate the nature of the singularities (0, 0, 1) and (1, 0, 0), equations (8) and (10) may be written as

d = A dx , ~.l ,jr

where p ' = p - p s , ' r ' = r - r s , a ' = c~-ots and p', a ' , r ' ~ 1, and

t A =- 1 /L G / L ~ "

From the roots of the equation,

I A - A I ] = O .

A necessary and sufficient condition for the existence of an exceptional curve through the singular points is found to be

i.e. the condition that the singularity be a saddle with only one negative root. The characteristic equation is a cubic of the form

A 3 - SI~ 2-}- D A - P =0,

where

S = G(1 + I l L )

D = G21L + ~

s]


Of course, S is the sum of the roots , D the sum of the p roduc t s of pairs of roots and P the p roduc t of the roots. P < 0 implies either one root < 0 or three roots < 0. S > 0 rules out the latter possibil i ty, ... P < 0, S > 0 implies one and only one negat ive root.

In turn, this implies one t ra jec tory only going into the singulari ty at (1,0,0) as x ~ . There fore , using equa t ion (10), we find at the final singularity

I='-~T + dr -~d < -~r)l -

and hence, with 0 < L ~< 1 and condi t ion (20), we have

and

~- + a - 1 i> O. (22)

Consider the values of (dp/dr) and (da/dr) at the singularities: f rom (14) at r = 0

~ ] d--rT o - 2

... For

o

G~ > G2 =) o > \ dr/o > 0. (23)

From (15), at r = 1

and, by (21),

dp) = G - V'[G 2 - 4(d4~/d~-),]

(dp2~ {dp~ < 0. (24) G1 > G 2 ~ \ d r ] l < \ d Z ] l

~iluo!o!:lJns aoj 'N u! leql ~ldtu! (g~) pue (lr~) suo!l!puo3 uoql 'I >> 9 o.loqnx '(I '9 -I) ~- N "o'! 'I = * jo pooqanoqq$.tou pueq 1JOl 13 oq N loI oax JI

(;Z) "0 > \,~p} > V~p}

(~rt_7)(,a_7) '/*p~ '[*p'~ ° 0 < (~# - '#)(I - 7) = V~P} - \ /

.(~t _,-i) (~ -7)

oS

--+I-=--

'0 > ~r/> tr/pu¢ I > 7 O3UlS

I - W- = \ 7op/ "."

D

7 ,7 '[*p_t I \op

"-/ '=*(,p/alp) 7 '(*P~ [ l=*(.~p/x)p) + I ~ = \ x)p]

oloq~

OA13q OAX '[ = .t Aj,!aeln~u!s oql 1~ olna s,Iel!d9H,l 3U!Aldde pue

"/ dr/ *p [ (l - 7o + *)9 = 7o--~

"0 > zr/> 'r/

s! (OI) uo!lenbo mON

(IRE) tuoaj '~iaeOlD

= tr/

o:l!aa~ sn lo'-I

.~=*(*p/Zdp) _ zr/,~=~(.~pfldp)

~D ~D

l~g SNOLLVflOH ~tAVA~ HV,qNI"INON dO SNOI,LITIOSNHDIH

332

small e, we have

B. F. GRAY AND M. E. SHERRINGTON

p 2 > p i > 0

a 2 > a l > O . (26)

and C2 as we proceed from

0 / 1 < ~ 2.

Suppose that C~ and C2 pass in 7r at point D. At D,

r = rD, p , ( r o ) = p ~ ( r o ) = p o .

By (15)

4,(ro, a2) > 4,(ro, al).

.'., using (8),

(dpq _ (dp2] 6 ( ~ D , - , ) - 6(~D, ~2) > 0. dr]o \d r }o=G1-G2- Po

Hence, applying the Intersect ion Theorem gives a contradiction with the values of (dp/dr) at r = 1.

(ii) C1 and C2 cannot first have a pass point in ~rz. Since there has been no pass point in ~-~ we have

p 2 > P l > O

and so

G___~ > G___~2 > 0. Pl /92

Suppose that G~ and G2 pass in "/7" 2 at point D. Then for L < 1,

rO+aD-- 1 >0 .

Consider the behaviour of trajectories C1 (1, 0, 0) towards (0, 0, 1).

(i) C1 and C2 cannot first have a pass point in ~-1. Since there has been no pass point in 7rz we still have the condition

E I G E N S O L U T I O N S O F N O N L I N E A R W A V E E Q U A T I O N S 333

By (10),

d r ] D \ dZ / D = L -~ -~2 >0.

So, applying the Intersection Theorem we have a contradiction with the values of (dp/d.c) at ~-= l(equation (25)).

(iii) C1 and C2 cannot first have an intersection point. Suppose that a point D was such a point. Then

'r = 'ro, p l = p2 = p o

O~ 1 -= O~ 2 = OlD.

We have

6(r~,, C~l) = 6 ( t o , a:) = 6 ( t o , no)

and so

(dpl d r } o \ d ~ ' / D

which again contradicts the values of (dp/d~-) at r = 1. Therefore, we deduce that the eigensolutions C, and C2 cannot cross

within the interval ~-~ (0, 1), and in particular at the point ~-= 0. This would require

(dpl _ < o, dz]o \d~-]o

which is a contradiction with the actual slopes there. This type of argument, which guarantees the uniqueness of the eigen-

value under these conditions, may be considered as a simple example of a form of 'courtesy principle'. That is, where the projections of two trajectories of an (n + 1) state space E in any plane ~'k can be shown to have no common point unless it has been preceded by an intersection in at least one of the other ( n - l) 7r-planes. Hence the impossibility of intersection in all the 7r-planes and, therefore, in the E state space itself.


6. Extension to More Than Two Variables. For a simple reaction in- volving more than one reactant species, application of the method is possible with no additional refinement. However , the process becomes progressively more difficult to visualise and we will briefly indicate this in the only other important case, a second order process:

A + B > products.

In this case, the equations and boundary conditions of our system become

d2r d~- dx 2 - G dx - ~;b(r, a,/3)

L d2a da A d - - ~ - G ~ - = ~b('r, a , f l ) (27)

d2/3 df l _ LB -d--Z- G d---~ - ch(T, a, [3),

where

x = - o ~ , r = 0 , a =/3 = 1 d r da d/3 dx - dx - dx = 0,

and

x = + ~ , r = l , a = / 3 = 0 d r _ da d/3 dx dx - d---x = 0,

and where LA, LB represent dimensionless transport coefficients of A and B respectively, and we restrict ourselves to 0 < LA ~ 1 , 0 < LB ~< 1.

If we consider the projections of two eigensolutions C~ and C2 of Z state space (7, p, a,/3) in the three planes,

7r,(p, r)plane 7rz(a, "r)plane rr3(/3, r)plane,

exactly analogous geometrical arguments show that a courtesy principle applies and in particular that the curves in 7r, have no common point. This produces a contradiction of the slopes at the cold singularity with those at the hot one, as before, thus implying the uniqueness of the eigensolution. The situation is illustrated in Fig. 5.


Figure 5.

"r=O ~-:l r = O "r=l

/ 3 = 1 ~

T : 0 -c=l

The behaviour of solutions in a three variable system if distinct eigenvalues were to exist.

7. Conclusions. We have shown that for an infinite one-dimensional system described by a set of nonlinear differential equations of the form (3), (4) or (27) there exists a discrete and unique eigenvalue or propagation velocity, provided that we require the quiescent medium in the region x ~ - w to be in a state which is asymptotically stable with respect to spatially homogeneous perturbations. The propagating wave-like solution is then independent of the small perturbations at the boundary, in contrast to solutions of the Fisher equation, which are unstable with respect to long wavelength perturbations. Another interesting equation describing propagating change in a nonlinear system, the cubic polynomial equation, has recently been discussed by Rosen (1980). In dimensionless form this single equation is in the present notation

d2~ - d~- - O d-~ + (~- - ~.2)(~. _ K) = 0, (28)

where

I > K > 0 ,

~(z) = (~" - ~.2)(~. _ K) (29)

and

thus falling into the class of equations discussed here, as distinct from the Fisher equation, for which the derivative at the singularity is zero. Equation (28) admits an exact analytical solution with eigenvalue (in the


s e n s e u s e d he re )

T h e a r g u m e n t s p r e s e n t e d h e r e s h o w this to be a u n i q u e e i g e n v a l u e f o r

this p a r t i c u l a r e q u a t i o n .

A P P E N D I X

To prove Theorem 1 of the text we need the generalised Mean Value Theorem, which states that for Yl and y2 continuous and differentiable on [0, c] there is a value, s r, 0 < s c < c, such that

y,(0) - y,(c) _ y'(~) y2(0) - y2(c) - y~(sc)"

In our case, x = 0 and x = c are assumed to be adjacent intersection points for yl(x), ydx), hence there exists a point ~, 0 < ~: < c where y',(~) = y~(~).

Application of the Mean Value Theorem for derivatives to the function g -= y~ - Y2 on the interval of 0 < s r' < ~, where g' > 0, gives us yj(s c) > y2(O.

Application of the same theorems on the interval [c, 0], with the assumption that y'l(c) > y~(c), leads us to conclude the existence of a point 3' such that y~(s r) = y~(~) and yl(y) < y2(7). We have thus shown the existence of two points in [0, c] such that

Y,(~) > Y2(~)

and

Y,(3') < Y2(3').

Hence, continuity implies the existence of a point 8 in [0, c] such that

yl(t~) = y2(t~)

and this contradicts our condition that the points x = 0 and x = c are adjacent crossing points.

Therefore, our assumption that y~(c) > y~(c) is incorrect. Therefore, we must have

y~(c) < y~(c),

as was to be proved.

T h e a u t h o r s w o u l d l ike to t h a n k t he r e f e r e e , Dr . N . F . B r i t t o n , f o r c o n s t r u c t i v e c r i t i c i s m o f t he o r i g i n a l m a n u s c r i p t .

L I T E R A T U R E

Frank-Kamenetskii, D. A. (1955). Diffusion and Heat Exchange in Chemical Kinetics. Princeton: University Press.


Kennerley, J. A. and J. Adler. (1966). "Stability of One-dimensional Laminar Flames with Distributed Heat Losses." Physics Fluids 9, 62-69.

Menkes, J. (1959). "On the Stability of a Plane Deflagration Wave." Proc. R. Soc. A253, 380-389.

Murray, J. D. (1977). Lectures on Nonlinear Differential Equation Models in Biology. Oxford: University Press.

Rosen, G. (1980). "On the Fisher and the Cubic-polynomial Equations for the Propagation of Species Properties." Bull. math. Biol. 42, 95-106.

Spalding, D. B. (1957). "I-D Laminar Flame Theory for Temperature Explicit Reaction Rates." Combust. Flame 1, 296-305.

RECEIVED 8-18-80 REVISED 6-22-81

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Eigensolutions of nonlinear wave equations in one dimension